Continuous Representations in the Statistical Theory of Electronic Energies

The use of continuous bases of representation other than a plane-wave basis is considered in the theory developed originally by Thomas and Fermi. The bases considered here are the sets of eigenfunctions of Hamiltonians corresponding to a particle subjected to a field of force which varies inversely as the cube of the distance to some fixed point. Variation of the strength of the interaction varies the basis continuously. Calculations of the energy of the hydrogen atom is carried out, with results that are appreciably closer to the quantum-mechanical result than is obtained with the original Thomas—Fermi theory. Numerical results suggest the possible existence of a statistical analog of the Rayleigh—Ritz variation principle.